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Chinese Remainder Approximation Theorem [article]

Matan Komisarchik
2016 arXiv   pre-print
We study a topological generalization of ideal co-maximality in topological rings and present some of its properties, including a generalization of the Chinese remainder theorem.  ...  Using the hyperspace uniformity, we prove a stronger version of this theorem concerning infinitely many ideals in supercomplete, pseudo-valuated rings. Finally we prove two interpolation theorems.  ...  Introduction This paper studies topological versions of the Chinese remainder theorem -CRT using two main concepts: topological co-maximality and the hyperspace uniformity.  ... 
arXiv:1509.02188v3 fatcat:bns4gbc35rayfas3gue47s6v2q

Around the Chinese Remainder Theorem [article]

Jean-Marie Didry, Pierre-Yves Gaillard
2008 arXiv   pre-print
We prove an explicit Chinese Remainder Theorem for one variable polynomials with complex coefficients, and derive some consequences.  ...  Chinese Remainder Theorem 3. f = D q + R.  ...  The polynomial R(X) in the Chinese Remainder Theorem (Theorem 8 page 6) is R(X) = D(a)=0 k+n<µa c a,k f (n) (a) n! (X − a) k+n , where c a,k is given by (4). Corollary 13.  ... 
arXiv:math/0412133v8 fatcat:n64ll73e3jbjjjmuwxenpbwmom

Robustness in Chinese Remainder Theorem [article]

Hanshen Xiao, Yufeng Huang, Yu Ye, Guoqiang Xiao
2017 arXiv   pre-print
Chinese Remainder Theorem (CRT) has been widely studied with its applications in frequency estimation, phase unwrapping, coding theory and distributed data storage.  ...  INTRODUCTION Conventional Chinese Remainder Theorem (CRT) is to reconstruct a single integer by its residues modulo several moduli.  ...  Theorem 5.  ... 
arXiv:1708.04089v3 fatcat:h2ajxxudvba5toqomlstv5pofu

The General Chinese Remainder Theorem

Oystein Ore
1952 The American mathematical monthly  
THE GENERAL CHINESE REMAINDER THEOREM OYSTEIN ORE, Yale University 1. Introduction. The Chinese remainder theorem, as one knows, is one of the most useful tools of elementary number theory.  ...  These products are evidently the symmetric functions Mi, M2, +++ so that we can state: sere sete wea a VW 1952] THE GENERAL CHINESE REMAINDER THEOREM 369 THEOREM 2.  ... 
doi:10.1080/00029890.1952.11988142 fatcat:5hjmvvkaoreu5d22fjsz74s66y

A density Chinese Remainder Theorem [article]

Jason Gibson
2013 arXiv   pre-print
Theorem 1.1 (Chinese Remainder Theorem). Let m 1 , . . . , m k be k positive integers, and let a 1 , . . . , a k be any k integers.  ...  Remainder Theorems.  ... 
arXiv:1302.0917v1 fatcat:v5pfz43govdqjgspziegmqrtee

A Multivariable Chinese Remainder Theorem [article]

Oliver Knill
2012 arXiv   pre-print
The Chinese remainder theorem is the special case, where A has only one column.  ...  Historical background The Chinese remainder theorem (CRT) is one of the oldest theorems in mathematics. It has been used to calculate calendars as early as the first century AD [5, 22] .  ...  The original paper of January 27, 2005 (google "multivariable chinese remainder") had been written in the context of multidimensional Diophantine approximation and was part of a talk on April 11, 2005  ... 
arXiv:1206.5114v1 fatcat:ubrhu7egx5g3jh3d3hqwpo72m4

The General Chinese Remainder Theorem

Oystein Ore
1952 The American mathematical monthly  
THE GENERAL CHINESE REMAINDER THEOREM OYSTEIN ORE, Yale University 1. Introduction. The Chinese remainder theorem, as one knows, is one of the most useful tools of elementary number theory.  ...  The remainder theorem. Let the system of congruences (1) be given where it is not assumed that the moduls are necessarily relatively prime.  ... 
doi:10.2307/2306804 fatcat:qlrc54qbznc7hggkfzvgrly2ha

On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors [article]

Guangwu Xu
2017 arXiv   pre-print
In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_  ...  This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo $4\tau$.  ...  Introduction The usual Chinese Remainder Theorem (CRT) concerns reconstructing an integer given its remainders with respect to a set of coprime moduli.  ... 
arXiv:1409.0121v4 fatcat:rjvrhdxhjbderlyju6zt5527cm

Multi-Stage Robust Chinese Remainder Theorem

Li Xiao, Xiang-Gen Xia, Wenjie Wang
2014 IEEE Transactions on Signal Processing  
It is well-known that the traditional Chinese remainder theorem (CRT) is not robust in the sense that a small error in a remainder may cause a large error in the reconstruction solution.  ...  In this special case, a closed-form reconstruction from erroneous remainders was proposed and a necessary and sufficient condition on the remainder errors was obtained.  ...  Multi-Stage Robust Chinese Remainder Theorem Li Xiao, Xiang-Gen Xia, Fellow, IEEE, and Wenjie Wang, Member, IEEE Abstract-It is well-known that the traditional Chinese remainder theorem (CRT) is not robust  ... 
doi:10.1109/tsp.2014.2339798 fatcat:jkslbxydvndtlhvwbnhkxjk66a

A Chinese Remainder Theorem for Partitions [article]

K. Seethalakshmi, Steven Spallone
2022 arXiv   pre-print
We present an analogue of the Chinese Remainder Theorem in this paper. Write C t for the set of t-cores.  ...  By the usual Chinese Remainder Theorem, we may view Z/mZ as the fibre product of Z/sZ and Z/tZ over Z/dZ. So we then investigate the effect of the functor S S k on a fibre product.  ...  We recall Theorem 3 from the Introduction. Theorem 3. If core d (σ) = core d (τ ), then there is a quasipolynomial Proof. Let i ≥ ℓ 0 .  ... 
arXiv:2105.07611v2 fatcat:kcdkntvdufhudlbcmkkdsdvrzy

Equidistribution from the Chinese Remainder Theorem [article]

Emmanuel Kowalski, Kannan Soundararajan
2020 arXiv   pre-print
We prove the equidistribution of subsets of (/)^n defined by fractional parts of subsets of (/q)^n that are constructed using the Chinese Remainder Theorem.  ...  These sets A q are built out of the sets A p v for prime powers p v using the Chinese Remainder Theorem.  ...  In this paper we show that Hooley's result may be recast as a general fact concerning the equidistribution of sets arising from the Chinese Remainder Theorem.  ... 
arXiv:2003.12965v2 fatcat:yjtwlkfvsfbevkxywto4hzjbui

Equidistribution from the Chinese Remainder Theorem

E. Kowalski, K. Soundararajan
2021 Advances in Mathematics  
These sets A q are built out of the sets A p v for prime powers p v using the Chinese Remainder Theorem.  ...  In this paper we show that Hooley's result may be recast as a general fact concerning the equidistribution of sets arising from the Chinese Remainder Theorem.  ...  By the Chinese Remainder Theorem α(x) is bounded by the product over p v q of the number of solutions to h · x = h · y (mod p v ), and this may be bounded by (p v ) if h ≡ 0 (mod p v ) and the resulting  ... 
doi:10.1016/j.aim.2021.107776 fatcat:dowri2q7oja6vbqpm7cdgetr54

Chinese remainder theorems and Galois modules

Neil Ormerod
1986 Journal of the Australian Mathematical Society  
In particular, the paper seeks to establish a relationship between Galois submodules of L and certain types of Chinese Remainder Theorems. 1980 Mathematics subject classification (Amer. Math.  ...  By the Chinese Remainder Theorem for L we have K V ® K L= 0 L w (cf. [2, pages 54 ff]).  ...  I 2 1 Chinese Remainder Theorems and Galois Modules 277 THEOREM Let L be a finite normal extension of K, with K a finite algebraic extension of Q, and let G = G(L/K).  ... 
doi:10.1017/s1446788700027257 fatcat:6dgpedrhofgpdii64d62qsa4b4

Poisson statistics via the Chinese remainder theorem [article]

A. Granville, P. Kurlberg
2005 arXiv   pre-print
We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem.  ...  We also study the spacings of subsets of Z/q_1q_2Z that are created via the Chinese remainder theorem from subsets of Z/q_1Z and Z/q_2Z (for q_1,q_2 coprime), and give criteria for when the spacings modulo  ...  Here we give several examples to indicate when we cannot expect some kind of "Central limit theorem" for the Chinese remainder theorem! 6.1. Counterexample 1.  ... 
arXiv:math/0412135v2 fatcat:yxwixjmksvb4neou55k2lec5ni

Computing Hilbert class polynomials with the Chinese remainder theorem

Andrew V. Sutherland
2010 Mathematics of Computation  
We present a space-efficient algorithm to compute the Hilbert class polynomial H_D(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem.  ...  Our algorithm is based on the CRT approach [1, 5, 17] , which computes the coefficients of H D modulo many "small" primes p and then applies the Chinese Remainder Theorem (CRT).  ...  We now suppose we have computed H D modulo primes p 1 , . . . , p n and consider how to compute H D mod P for an arbitrary positive integer P , using the Chinese Remainder Theorem.  ... 
doi:10.1090/s0025-5718-2010-02373-7 fatcat:bthdblwvvzgofhyrigrz46oowm
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